Given an array A of non-negative integers, the array is squareful if for every pair of adjacent elements, their sum is a perfect square.

Return the number of permutations of A that are squareful. Two permutations A1 and A2 differ if and only if there is some index i such that A1[i] != A2[i].

Example 1:

Output: 2
Explanation: 
[1,8,17] and [17,8,1] are the valid permutations.

Example 2:

Input: [2,2,2]
Output: 1

Note:

1 <= A.length <= 12
0 <= A[i] <= 1e9

Solved this with PHP.

1. Go through the input and construct array ‘peers’
   1. peers holds an array of pointers pointing to all possible ‘partners’ in the input.
   2. For example, [1 => [8=>ref(8)], 8 => [1=>ref(1), 17=>ref(17)], 17 => [8=>ref(8)]]
2. Meanwhile, another array stat collect all numbers and its occurrence.
3. Then peers is recursive. so it’s like a tree, with infinite depth.
4. Starting from the root of peers, if found any number in ‘stat’, update (-1) stat, and move on to peers’ children
   1. if cannot find any match at some points, break;
   2. since stat is finite, the recursive func will run forever
5. Calculate the sum.

Works well with even larger input length.

16ms runtime w/ 14.7M RAM

Class Solution {

    /**
     * @param Integer[] $ls
     * @return Integer
     */
    function numSquarefulPerms($ls) {
        
        if(count($ls) <= 1){
            return 0;
        }
        
        $peers = [];
        $stat = [];
        
        for($i = 0; $i < count($ls); $i++){
            // Insert/Update stat
            if(!isset($stat[$ls[$i]])){
                $stat[$ls[$i]] = 1;
            } else {
                $stat[$ls[$i]] ++;
            }
            
            for($j = $i + 1; $j < count($ls); $j++){

                $value_i = $ls[$i];
                $value_j = $ls[$j];
                
                if(!isset($peers[$value_i])){
                    $peers[$value_i] = [];
                }
                if(!isset($peers[$value_j])){
                    $peers[$value_j] = [];
                }
                if($this -> isPerfectSquare($ls[$i] + $ls[$j])){
                    $peers[$value_i][$value_j] = &$peers[$value_j];
                    if($ls[$j] !== $ls[$i]){
                        $peers[$value_j][$value_i] = &$peers[$value_i];
                    }
                }
            }
        }
        // If any number has no partner
        foreach($peers as $p){
            if($p === []){
                return 0;
            }
        }
        return $this->traversal($stat, $peers, count($ls));
    }
    
    // Check if number is perfect square w/ small optimization
    function isPerfectSquare($num){
        $last_digit = $num % 10;
        if($last_digit == 2 ||
            $last_digit == 3 ||
            $last_digit == 7 ||
            $last_digit == 8 
          ) 
            return false;
        
        $sqrt = sqrt($num);
        return $sqrt == floor($sqrt);
    }
    
    function traversal($nums, &$tree, $max){
        $count = 0;
        var_dump($nums);
        if(empty($nums)){
            echo "Add\r\n";
            return 1;
        }
        foreach($tree as $k => $v){
            // Make a copy
            $nums_copy = $nums;
            if(isset($nums_copy[$k])){
                // If exist decrease, if last remove
                if($nums[$k] > 1){
                    $nums_copy[$k] --;
                } else {
                    unset($nums_copy[$k]);
                }
                // move on to children
                $count += $this -> traversal($nums_copy, $v, $max - 1); 
            }     
        }
        return $count;
    }
}